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By W. W. Rouse Ball

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1 for d = 3, for which this covering result may not be sufficient. 8c. Tangent field to a null set. We consider here a possible generalization of the notion of weak tangent field to higher dimension. Let E be a set in Rd , τ a map from E into the Grassmann manifold G(d, d − 1) of hyperplanes in Rd , and k an integer between 1 and d − 1. e. x ∈ S ∩ E. 4, we can show that every null sets in R3 admits a 2-weak tangent field, but we do not know if every null set in R3 admits a 1-weak tangent field. 5 were given a positive answer.

1, and a careful truncation-and-localization argument. The price to pay is that the resulting function f could still be differentiable at some point of E. 1 in the positive. The construction required for the strong formulation is considerably more complicated. 8. 1, and of other results from the previous sections. Since many relevant questions are still unanswered even in dimension three, the following discussion will be sometimes restricted to this case. 8a. Covering of finite sets. As usual, x, y, z denote the coordinates of points in the space.

Ambrosio, N. Fusco, D. Pallara: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. Oxford Science Publications, Oxford, 1999. u. set. u. u. subset. The uniqueness of τ follows by the existence of f¯. 16If not, we could find a curve S of class C 1 and a Lipschitz map f such that f is not differentiable in the direction τS for a subset with positive length of S∩E, and this contradicts Rademacher theorem in dimension 1. 22 G. Alberti, M. Cs¨ ornyei and D.

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